Suppose \(\gamma <0\) , which implies that \(\theta _F= U(\infty )<\infty \)  and that \(a_N = \theta _F- F^{-1}(1-1/N)=U_F(\infty ) - U_F(N)\) . By corollary 1.2.10 in [1]}, \( (U_F(\infty )- U(x))^{-1}\in RV_{-\gamma }\)  and we can write \(1/(U_F(\infty )- U(x))= x^{-\gamma }L(x)\)  for some slowly varying function \(L\)  by lemma REF . In particular, \(1/a_{N} =1/(U_F(\infty )- U(N)) = N^{-\gamma }L(N) \) .
The \(F\)  terms satisfy
\(& \dfrac{1}{a_N}\left(F^{-1}\left(1- \frac{1}{N\tau } \frac{\log (T)}{\log (T)+1} \right) - F^{-1}\left(1- \dfrac{1}{N\tau } \right)\right)\\& = \dfrac{1}{a_N}\left(U_F\left(N\tau \dfrac{\log (T)+1}{\log (T)} \right)- U_F(N\tau ) \right)\\&= \dfrac{1}{U_F(\infty )-U_F(N)}\left( \left( U_F\left(N\tau \dfrac{\log (T)+1}{\log (T)} \right)- U_F(\infty ) \right)- ( U_F(N\tau )- U(\infty )) \right)\\& = N^{-\gamma }L(N)\left( \left(N\tau \dfrac{\log (T)+1}{\log (T)} \right)^{\gamma } \dfrac{1}{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - (N\tau )^{\gamma } \dfrac{1}{L(N\tau )} \right)\\& \propto \left( \dfrac{\log (T)+1}{\log (T)} \right)^{\gamma } \dfrac{L(N) }{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - \dfrac{L(N)}{L(N\tau )} \\& \rightarrow 0,\) 
